Ah, I see now. No, the symmetry group is understood to act locally at each point, on the tangent space at that point. Thus there is a group of transformations ##S## at each point, with the group structure set by the metric ##g## at that point. The singular value decomposition, etc., can all be...
I appreciate the suggestion, but I'm not sure that specific examples of the decomposition ##g = UDU^T## are particularly illuminating, even in two dimensions. This decomposition is also known as the singular value decomposition of a symmetric real matrix and can be computed by essentially any...
Thanks for the good question.
For the non-trivial solutions to the defining condition
$$
g = S^T g S ,
$$
suppose first that ##g## has signature ##(+,+,+,+)##. We can use the eigenvalue decomposition to write ##g = UDU^T## where ##U## is orthogonal and ##D## is diagonal with positive entries...
For a general spacetime metric ##g##, consider the set of transformations ##S## on ##\mathbb{R}^4## such that
$$
g(u,v) = g(Su, Sv)
$$
for all ##u,v##. The set of such transformations satisfies the group axioms. Maybe I gave it the wrong name, but this group is what I meant by the symmetry group...
No, I defined this operation on all complex 2-by-2 matrices, and primarily used it on the matrices ##e_i##. As described in the original post, the ##e_i## are 2-by-2 Hermitian and characterised by the condition
$$
g_{ij} = \frac{1}{2}( e_i \bar{e_j} + e_j \bar{e_i} ) .
$$
For a general spacetime...
I'd like to better understand spinors on curved spacetime, but started wandering along the following tangent. I've looked at but not particularly understood the sections on spinors in the texts by Penrose and (Misner, Thorne and Wheeler).
Let ##g_{ij}## be a spacetime metric (a symmetric...
There may be some useful comments amongst this fairly ancient thread
https://www.physicsforums.com/showthread.php?t=217846
(probably my own posts in it were too terse to be much help.) There were some references mentioned which might be more helpful. Bogoliubov & Shirkov definitely does...
Interesting. Personally, I'm a fan of canonical quantization because I think it is easier to identify the underlying hypotheses of the theory.
I vaguely recall that some of the early proofs in non-abelian QFT were first constructed using path integrals -- Slavnov-Taylor identities & BRST...
Interesting. Personally, I'm a fan of canonical quantization because I think it is easier to identify the underlying hypotheses of the theory.
I vaguely recall that some of the early proofs in non-abelian QFT were first constructed using path integrals -- Slavnov-Taylor identities & BRST...
You can't Lorentz transform a time-like interval into a space-like interval. The Minkowski norm
t^2 - x^2 - y^2 - z^2
is invariant under Lorentz transformations, so a time-like interval (positive norm) cannot be mapped into a space-like interval (negative norm).
I disagree on all points:
quantum thermal states live in Fock space just as much as a classical microstate lives in classical configuration space.
I'm still getting to grips with the finer points of Glimm and Jaffe (thanks A. Neumaier), but without doubt, Fock space is still the starting...
Ah, then it's likely I didn't actually understand Haag's theorem. I will try to look into it again. Could you comment on whether Haag's theorem is consistent with the work of Glimm and Jaffe in 1+1 and 2+1 dimensions? These authors explicitly construct an interacting relativistic field theory...
Each of the p_k are in R^3, and take psi in the temporal gauge (0-component vanishes). Given the Fourier transform is well-defined for all square-integrable functions \psi_N(p_1,...,p_N), doesn't the transversality condition merely restrict physical states for the photon to a proper subspace...
The N-particle wave function \psi_N(p_1,...,p_N) describes an amplitude for N particles with momenta p_1, ... p_N. Could we not simply define a position representation \psi_N(x_1,...,x_N) by taking the Fourier transform with respect to each p_j? (Square integrability of either guarantees...